Page 125 - Reservoir Geomechanics

109 Rock failure in compression, tension and shear

Nearly all proposed formulae for determination of rock strength from geophysical
logs utilize either:
P-wave velocity, V p ,or equivalently, the travel time of compressional waves along

−1
the wellbore wall,
t (
t = V p ), expressed as slowness, typically µs/ft,
Young’s modulus, E (usually derived from V p and density data as illustrated in Table

3.1), or
Porosity, φ (or density) data.

The justification for proposed relations is illustrated by the dependence of uniaxial
compressive strength on these parameters as shown in Figures 4.14, 4.15 and 4.16 for
sandstones, shales, and limestone and dolomite, respectively utilizing data from Lama
and Vutukuri (1978), Carmichael (1982), Jizba (1991), Wong, David et al.(1997),
Horsrud (2001) and Kwasniewski (1989). In each of the figures, the origin of the
laboratory data is indicated by the symbol used. Despite the considerable scatter in
the data, for each rock type, there is marked increase of strength with V p and E and a
marked decrease in strength with increased porosity.
Table 4.1 presents a number of relations for different sandstones from different geo-
logical settings for predicting rock strength from log data studied (Chang, Zoback
et al. 2006). Equations (1)–(3) use P-wave velocity, V p (or equivalently as
t) mea-
surements obtained from well logs. Equations (5)–(7) utilize both density and V p data,
and equation (4) utilizes V p , density, Poisson’s ratio (requiring V s measurements) and
clay volume (from gamma ray logs). Equation (8) utilizes Young’s modulus, E, derived
from V p and V s , and equations (9) and (10) utilize log-derived porosity measurements to
estimate rock strength. Because of the considerable scatter in Figure 4.14,itisobvious
that it would be impossible for any of the relations in Table 4.1 to fit all of the data
shown. It also needs to be remembered that P-wave velocity in the lab is measured at
ultrasonic frequencies (typically ∼1 MHz) in a direction that is frequently orthogonal to
bedding and typically on the most intact samples available that may not be representa-
tive of weaker rocks responsibility for wellbore failure. As discussed in Chapter 3, sonic
velocities used in geophysical well logs operate at much lower frequencies. Moreover,
such measurements are made parallel to the wellbore axis, which frequently is not per-
pendicular to bedding. Hence, there are significant experimental differences between
field measurements and laboratory calibrations that need to be taken into consideration.
For the relations that are based on V p (Figure 4.14a), it is noteworthy that except for
equations (1) and (6) (derived for relatively strong rocks), all of the relations predict
extremely low strengths for very slow velocities, or high travel times (
t ≥ 100 µs/ft),
and appear to badly underpredict the data. Such velocities are characteristic of weak
sands such as found in the Gulf of Mexico, and a number of the relations in Table 4.1
were derived for these data. However, while equation (6) appears to fit the reported
values better than the other equations, one needs to keep in mind that there are essen-
tially no very weak GOM sands represented in the strength data available in the three
studies represented in this compilation. Similarly, for fast, high-strength rocks, equation